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From Equation (6), $B(t,T)=-t+c(T)$ for some function $c(T)$. $1=P(t,t)=e^{-A(t,t)-(c(t)-t)r_t}$ or $A(t,t)+(c(t)-t)r_t=0,\,\forall (r_t,t)$. So $c(t)=t, A(t,t)=0,\forall t$. For Equation (8) you have missed the square on $\sigma$ and a factor of $\frac13$. Then you just need to substitute in the function for $b(s)$ and integrate the following to get the ...

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[Question 1] Let us define \begin{align} X_t &= X_0 \exp((r_d-r_f-\frac{1}{2}\sigma^2)t + \sigma W_t) \\ &= X_0 \exp((r_d-r_f)t) \mathcal{E}(\sigma W_t) \end{align} then, in that case $$E(X_t \vert \mathcal{F}_0) = X_0 \exp((r_d-r_f)t) = F^X(0,t)$$ only because $$\mathcal{E}(\sigma W_t)$$ is a stochastic exponential (strictly positive martingale ...

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For starters, the short rate model you mention in equation (1) is Cox-Ingersoll-Ross while the bond price in equations (2)-(4) correspond to the Vacisek model. So there is a problem somewhere, I would go for a typo in (1). Second, what you wrote seems fine to me, so there must definitely be yet another typo in your solution manual. Note that if there is no ...

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Assume deterministic and constant interest rates. For an investor in the foreign economy i.e. a market participant that can only trade assets delivering a payout in the foreign currency, let us define $$\tilde{X}_t = \tilde{X}_0 \exp \left(\left(r_f-r_d-\frac{\sigma_\tilde{X}^2}{2}\right)+\sigma_\tilde{X} W_t^{\tilde{X},\mathbb{Q}^f} \right)$$ $$Y_t ... 1 Just use the fact that$$ \sigma_X W_t^1 + \sigma_Y W_t^2 = \sqrt{ \sigma_X^2 + \sigma_Y^2 + 2\rho\sigma_X\sigma_Y } W_t $$holds in probability assuming that W_t^1 and W_t^2 are 2 correlated Brownian motions with$$ d\langle W_t^1, W_t^2 \rangle_t = \rho dt $$and W_t is a new standard Brownian motion defined over the same probability space. Simply ... 1 Equations (1) to (3) are correct. Your investment strategy is then, \forall t > 0$$ X_t = \theta _ t S_t $$Provided you use this strategy as part of self-financing portfolio you can write the P&L over an infinitesimal time interval as$$ dV_t = \theta_ t dS_t  assuming zero safe rate, i.e. that any cash required to finance your long stock ...

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